Question

If asecθ - btanθ = 5 and atanθ - bsecθ = 3; then find the value of (a² - b²)/4.
1. 4
2. - 4
3. - 3 
4. 3

Answer 1

Correct Answer - Option 1 : 4

Given:

asecθ - btanθ = 5      ----(i)

atanθ - bsecθ = 3      ----(ii)

Formula used:

sec²θ - tan²θ = 1

(a - b)² = a² + b² - 2ab

Calculation:

asecθ - btanθ = 5      ----(i)

atanθ - bsecθ = 3      ----(ii)

Squaring both side we get from (i),

a²sec²θ + b²tan²θ - 2absecθtanθ = 25      ----(iii)

Squaring both side we get from (ii),

a2tan2θ + b2sec2θ - 2absecθtanθ = 9      ----(iv)

Now subtracting equation (iv) from equation (iii), we get;

a2sec2θ + b2tan2θ - a2tan2θ - b2sec2θ = 16

⇒ a²(sec²θ - tan²θ ) - b2(sec2θ - tan2θ ) = 16

⇒ a² - b² = 16

Now,

(a2 - b2)/4 

⇒ 16/4

⇒ 4

∴ The required value is 4.